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So why come back to this numerical lecture in the previous lectures?

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I wanted to show you that you can really solve the stationary shooting equation for some examples.

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And what we have used is the so-called shooting method, and you have seen that it's actually quite

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delicate and quite unstable because we always had to find the perfect art, some some some good value

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for our propagation border that we called a.

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So if a was too large, then this exponential behavior of a whole wave function became a real problem

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so that the solution exploded.

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And this was basically because we have used a very simple approach for calculating derivatives.

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We have just used the old value, plus some delta of the wave function, for example.

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And of course, they are much, much better ways to do this.

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For example, you can use a whole McWhirter algorithm to propagate the wave function.

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If you would do this, the whole shooting method would work much better, and you wouldn't have to be

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so careful about the choice of eight.

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So this is something that you can do on your own if you want just Google, only Qatar.

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But also, here's something that I want to show you.

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So in case you have access to Mathematica.

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But it has to be version ten point two or higher.

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You can use the command called and the EIGEN system so he can see all the commands that are needed to

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calculate the EIGEN system of the harmonic oscillator, as we have done previously.

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So we again define some value h just our Planck constant bar.

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We defined the potential which is basically x squared times a prefecture.

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And this is our our tweeting equation.

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And then we just solve the eigen spectrum of this Hamiltonian.

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So actually, we don't really have the shooting equation.

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We only have the Hamiltonian.

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And then there is there, there's a command and the eigen system that just determines the eigenvalues

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and diegan functions of this Hamiltonian.

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So of course, when you used his methods, you don't really learn anything about the methods.

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So you don't know how the computer solves this problem, but you get to solutions very, very fast.

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And as you can see, the ion energies are again 0.5, 1.5 and so on.

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So it just increments of one as we have previously.

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And you can also see here we can plot these wave functions on top of the eigenvalues.

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So this is a much simpler and much faster way to solve such problems.

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But of course, I think it was worth the effort to show you that you can also do it by yourself and

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Python using really basic concepts like using the most easy method for Celtic calculating the derivative.

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So as I told you, this is not really very, not really stable, and you have to be very careful with

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the choice of some parameters.

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But the purpose of this lecture was just to show you that it's still possible.

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Now, when you want to become more professional, you have to find more elaborate algorithms or you

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could use tools like this.